| L(s) = 1 | + (−3.95e4 + 2.59e5i)2-s − 1.35e8i·3-s + (−6.55e10 − 2.04e10i)4-s − 1.24e12·5-s + (3.51e13 + 5.36e12i)6-s + 4.20e14i·7-s + (7.90e15 − 1.61e16i)8-s + 1.31e17·9-s + (4.90e16 − 3.21e17i)10-s − 3.22e18i·11-s + (−2.77e18 + 8.89e18i)12-s − 2.28e19·13-s + (−1.08e20 − 1.66e19i)14-s + 1.68e20i·15-s + (3.88e21 + 2.68e21i)16-s + 6.53e21·17-s + ⋯ |
| L(s) = 1 | + (−0.150 + 0.988i)2-s − 0.350i·3-s + (−0.954 − 0.298i)4-s − 0.325·5-s + (0.346 + 0.0527i)6-s + 0.258i·7-s + (0.438 − 0.898i)8-s + 0.877·9-s + (0.0490 − 0.321i)10-s − 0.579i·11-s + (−0.104 + 0.334i)12-s − 0.203·13-s + (−0.255 − 0.0389i)14-s + 0.113i·15-s + (0.822 + 0.569i)16-s + 0.464·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.954 - 0.298i)\, \overline{\Lambda}(37-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+18) \, L(s)\cr =\mathstrut & (-0.954 - 0.298i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{37}{2})\) |
\(\approx\) |
\(0.8485647273\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8485647273\) |
| \(L(19)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (3.95e4 - 2.59e5i)T \) |
| good | 3 | \( 1 + 1.35e8iT - 1.50e17T^{2} \) |
| 5 | \( 1 + 1.24e12T + 1.45e25T^{2} \) |
| 7 | \( 1 - 4.20e14iT - 2.65e30T^{2} \) |
| 11 | \( 1 + 3.22e18iT - 3.09e37T^{2} \) |
| 13 | \( 1 + 2.28e19T + 1.26e40T^{2} \) |
| 17 | \( 1 - 6.53e21T + 1.97e44T^{2} \) |
| 19 | \( 1 - 1.34e23iT - 1.08e46T^{2} \) |
| 23 | \( 1 + 4.96e22iT - 1.05e49T^{2} \) |
| 29 | \( 1 + 1.19e26T + 4.42e52T^{2} \) |
| 31 | \( 1 - 7.37e26iT - 4.88e53T^{2} \) |
| 37 | \( 1 + 2.48e28T + 2.85e56T^{2} \) |
| 41 | \( 1 + 1.04e29T + 1.14e58T^{2} \) |
| 43 | \( 1 - 2.34e29iT - 6.38e58T^{2} \) |
| 47 | \( 1 + 6.54e29iT - 1.56e60T^{2} \) |
| 53 | \( 1 - 8.42e30T + 1.18e62T^{2} \) |
| 59 | \( 1 - 6.65e31iT - 5.63e63T^{2} \) |
| 61 | \( 1 + 1.74e32T + 1.87e64T^{2} \) |
| 67 | \( 1 - 1.02e33iT - 5.47e65T^{2} \) |
| 71 | \( 1 - 3.10e33iT - 4.41e66T^{2} \) |
| 73 | \( 1 - 3.37e33T + 1.20e67T^{2} \) |
| 79 | \( 1 - 1.58e34iT - 2.06e68T^{2} \) |
| 83 | \( 1 - 6.39e34iT - 1.22e69T^{2} \) |
| 89 | \( 1 + 1.63e35T + 1.50e70T^{2} \) |
| 97 | \( 1 - 3.39e35T + 3.34e71T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.61671313333665507998896060816, −15.39407515551796270418868289241, −13.88239859469266406811477417351, −12.35720465589958884882623186301, −10.05354839412489626708119923530, −8.341416828020293844337545739779, −7.09561509100021190180348718714, −5.57656958739945949398556852918, −3.84518782284908815046630527647, −1.33491121188158257291189230602,
0.29639372819466899118334388165, 1.89237082299382099286779909451, 3.64446384261646038410972452633, 4.82731025156185558661027010665, 7.47310374492983621997322779449, 9.333295384455297205278226166733, 10.53043374575063560442399056725, 12.04985125254867634496149951364, 13.46210271741980523109952164476, 15.33879042723034974970817073931
